My Favorite One-Liners: Part 6

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Sometimes, I’ll expect students to learn and master operations that cancel. For example, in Precalculus, I want my students to know the sum-to-product trigonometric identities

\sin u + \sin v = \displaystyle 2 \sin\left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right),

\sin u - \sin v = \displaystyle 2 \cos\left( \frac{u+v}{2} \right) \sin\left( \frac{u-v}{2} \right),

\cos u + \cos v = \displaystyle 2 \cos\left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right),

\cos u - \cos v = \displaystyle -2 \sin \left( \frac{u+v}{2} \right) \sin\left( \frac{u-v}{2} \right).

These can be helpful for solving trigonometric equations. For example, to solve \cos 3x + \cos 7x = 0, we have

\cos 3x + \cos 7x = 0

\displaystyle 2 \sin\left( \frac{3x+7x}{2} \right) \cos \left( \frac{3x-7x}{2} \right) = 0

2 \sin 5x \cos (-2x) = 0

2 \sin 5x \cos 2x = 0

\displaystyle x = \frac{n\pi}{5} \qquad  \hbox{or} \qquad \displaystyle x = \left( \frac{n}{2} + \frac{1}{4} \right)\pi for integers n.

However, I also want my students to know the product-to-sum trigonometric identities

\cos x \cos y = \displaystyle \frac{1}{2} [\cos(x+y) + \cos(x-y) ],

\sin x \sin y = \displaystyle \frac{1}{2} [\cos(x-y) - \cos(x+y) ],

\sin x \cos y = \displaystyle \frac{1}{2} [\sin(x+y) + \sin(x-y) ].

These are useful when computing certain definite integrals (especially related to Fourier series). For example, if m \ne n are both integers, then

\displaystyle \int_0^{2\pi} \cos mx \cos nx \, dx = \displaystyle \int_0^{2\pi} \frac{1}{2} \left[\cos([m+n]x) + \cos([m-n])x) \right] \, dx

= \left[ \displaystyle \frac{\sin([m+n]x)}{2(m+n)} +  \frac{\sin([m-n]x)}{2(m-n)} \right]^{2\pi}_0

= \displaystyle \frac{\sin(2[m+n]\pi) - \sin 0}{2(m+n)} +  \frac{\sin(2[m-n]\pi) - \sin 0}{2(m-n)}.


This integral and other similar integrals are necessary to find the formula for the coefficients in a Fourier series.

In other words, sometimes I’ll want my students to convert a product into a sum. Other times, I’ll want my students to convert a sum into a product.

To help this sink in, I’ll tell my students, “To quote the great philosopher: Sometimes you gotta know when to hold ’em, know when to fold them.”

However, when I made this joke recently, a student innocently asked, “What great philosopher said that?” I turned the question back to my class, but not one of my class of millennials knew the answer. One person came close with his answer of “Willie” — wrong answer but correct genre and time frame. (Somebody else answered Socrates.)

So that my students actually learn something important in my class, here’s the cultural reference:

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1 Comment

  1. Sometimes I think my best moments teaching mathematics are when I digress into some peculiar bit of culture. It wakes students up wondering how their Numerical Methods course jumped from Runge-Kutta methods over to how stores came to have set, fixed prices on tags and everything.


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